Slides from the lecture - SigmaCamp

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14 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

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Bernoulli’s law

and

Magnus force


Blaise

Pascal

P

=
ρ
gh


P

=
ρ
gh




Pressure in liquid/gas is isotropic. It acts equally in all directions



Pressure is force per unit area



Due to the gravity, pressure at a given level equals to the weight
of the column of liquid/gas above this level over a unit area

ρ=
fluid/gas density

g
=acceleration due to gravity

h
=height

P

=
ρ
gh




For a non
-
turbulent flow of fluid or gas



As speed increases, the pressure in the fluid or gas decreases.

P + ½
ρ
v
2
+
ρ
gh

= const


P=pressure of the fluid/gas along the streamline

v
=velocity of the fluid/gas along the streamline

g
=acceleration due to gravity

h
=height

ρ=
fluid/gas density

The Bernoulli’s equation expresses conservation of
enegy
.
It assumes that:

The fluid/gas has a constant density

The fluid/gas is traveling in a steady flow

There is no friction

The fluid/gas
is non viscous and
incompressable

Acceleration

a

-
a

Acceleration in the
non
-
inertial frame
moving with the flow

Because velocity of the fluid/gas flow has changed (increased)
from
v
1

to
v
2

, there must be a force which causes it to accelerate
while passing the distance
l
.

For simplicity, let us assume constant acceleration
a
.

Distance
l

Acceleration

a

-
a

Acceleration in the
non
-
inertial frame
moving with the flow

The equivalence principle
:

In an accelerated reference frame moving with the flow we can
calculate the pressure difference as if it were a pressure

difference in a gravitational field,
𝚫
P = P
2

-

P
1

=
ρ

a
l

Distance
l

The
inertial mass

relates force and acceleration in the
Newton’s
first law of motion
: F = m
a
.

The gravitational mass determines force of gravitational
attraction in the
Newton’s law of gravity
: (= m
g
).

The inertial mass and the gravitational mass are equal
.

Acceleration

a

-
a

Acceleration in the
non
-
inertial frame
moving with the flow

Kinematics of motion with constant acceleration,
a
, gives,

v
2

=
v
1

+
at
,

l

=
v
1
t
+ ½
at
2

= (
v
2
2

-

v
1
2

)

/(2
a
)

where
t

is the time it took the flow to pass the distance
l
.

Distance
l

Acceleration

a

-
a

Acceleration in the
non
-
inertial frame
moving with the flow

Combining the two results gives the Bernoulli equation,


𝚫
P = P
2

-

P
1

=
ρ

a
l
=
ρ

(
v
2
2

-

v
1
2

)/2

Distance
l

ρ
v
2
/2+P
atm
=

ρ
gh
+P
atm

=>

v
2

= 2
gh

P
atm

P
atm

Dental Saliva Ejector Hose With
Water
Venturi

Suction System

Ventouri

wine aerator

Ventouri

detergent intake
system in a
powerwasher

Becomes important for
wind velocity
v

> √2
gh

(≈ 10 m/s for h ≈ 5 m).

Ships sailing side by side can get too close together (as in picture above, at a certain
point during the refueling). When this happens, the
Venturi

effect takes over, and the
ships will head toward an unavoidable collision

An airfoil creates a region of high pressure air below the wing, and
a low pressure region above it. The air leaving the wing has a

downward flow creating the Newtonian force.

Bernoulli pressure
field creates the downwash.


Where the cylinder is turning into the airflow, the air is
moving faster and the pressure is lower


Where the cylinder is turning away from the airflow,
the air is moving slower and the pressure is greater


The cylinder moves towards the low pressure zone

The Magnus effect!



stitches help the ball to catch the air



the baseball curves towards the lower air pressure

Typical ball spin
-
rates are:

3,600 rpm when hit with a 10Â
°

driver (8Â
°

launch angle) at a velocity of 134 mph

7,200 rpm when hit with a 5 iron (23Â
°

launch
angle) at a velocity of 105 mph

10,800 rpm when hit with a 9 iron (45Â
°

launch
angle) at a velocity of 90 mph

Topping the ball (i.e. when the bottom of
the club
-
face hits the ball above its center)
will cause the ball to spin in the other
direction
-

i.e. downward
-

which will cause
the ball to dive into the ground.

Dimples cause the air
-
flow
above the ball to travel faster
and thus the pressure on the ball
from the top to be lower than
the air pressure below the ball.
This pressure difference (i.e.
more relative pressure from
below than on top) causes the
ball to lift (Magnus effect) and
stay in the air for a longer time.

1927